Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C29 · Schur multiplier (H2) = C25 |
| direct product | C29×#2760 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C29 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C29 |
| prime radical | C29×#392 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C29 ◁ G | |
520 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 520 conjugacy classes of subgroups — too many to draw. Only its 176 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1856 |
A minimal faithful permutation representation — 4 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29)(31,34)(32,36)(38,39,42,40)(41,44,45,43)(46,47)(48,52)(49,50)(51,53);(31,34)(35,37)(39,40)(41,45)(47,50)(48,53);(30,31,33,34)(32,35,36,37)(38,40,42,39)(41,44,45,43)(46,48)(47,51)(49,53)(50,52);(30,32)(31,35)(33,36)(34,37)(38,41)(39,43)(40,44)(42,45)(46,47)(48,51)(49,50)(52,53)928 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#12187, C29×#12263, C29×#36776, … |
| maximal normal subgroup of | C29×#12187, C29×#12263, C29×#36776, … |
| maximal quotient of | C29×#14375, C29×#14508, C29×#14572, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1492);